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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Perturbed dynamical systems with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations

Author: B. Perthame
Journal: Trans. Amer. Math. Soc. 317 (1990), 723-748
MSC: Primary 35B25; Secondary 35F99, 58F30
MathSciNet review: 943607
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Abstract: We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system

\begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ... \quad {\text{on}}\;\partial \Omega . \hfill \\ \end{gathered} \end{displaymath}

We prove that, if $ b$ has a single attractive singular point, $ {u_\varepsilon }$ converges uniformly on compact subsets of $ \Omega $, and with an exponential decay, to a constant $ \mu $, and we determine $ \mu $. We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations.

Résumé. Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d'un système dynamique:

\begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ...\quad {\text{sur}}\;\partial \Omega . \hfill \\ \end{gathered} \end{displaymath}

Nous prouvons que, si $ b$ a un seul point singulier attractif, alors $ {u_\varepsilon }$ converge vers une constant $ \mu $, uniformément sur tout compact, et avec une vitesse exponentielle. Nous déterminons $ \mu $. Nous traitons aussi le cas de conditions aux limites de Neuman. Pour cela, nous faisons l'analyse asymptotique d'une mesure ergodique intervenant naturellement dans le problème, ce qui revient à étudier la limite par viscosité évanescente dans une équation de Hamilton-Jacobi. Ceci est réalisé sous des hypothèses très générales gâce à un passage à la limite faible dans cette équation.

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Keywords: Dynamical systems, singular perturbations, Hamilton-Jacobi equations, viscosity limit, ergodic measure, large deviations
Article copyright: © Copyright 1990 American Mathematical Society

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